Mechanical effects of tunneling on adjacent pipelines based on Galerkin solution and layered transfer matrix solution

Soils and Foundations 2013;53(4):557–568 The Japanese Geotechnical Society

Soils and Foundations www.sciencedirect.com journal homepage: www.elsevier.com/locate/sandf

Mechanical effects of tunneling on adjacent pipelines based on Galerkin solution and layered transfer matrix solution Zhiguo Zhanga,b,c,d,n, Mengxi Zhangd a

School of Environment and Architecture, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China b State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China c Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China d Department of Civil Engineering, Shanghai University, Shanghai 200072, China Received 20 July 2012; received in revised form 13 April 2013; accepted 4 May 2013 Available online 24 July 2013

Abstract The mechanical analysis of undercrossing tunneling on adjacent existing pipelines is an important challenge that geotechnical engineers may need to face when designing new excavation projects. A Galerkin solution and a layered transfer matrix solution for the tunnel–soil–pipeline interaction are given in order to compare the effects of soil stratifications on the pipeline behavior subjected to tunnel-induced soil movements. For the Galerkin solution, the soil is modeled by the modulus of subgrade reaction and the governing differential equations are converted to finite element equations using the Galerkin method. To take full consideration for non-homogeneous soil characteristics, a layered soil model is employed in the layered transfer matrix solution by applying the double Laplace transform and transfer matrix method. The differences between the two proposed solutions are verified with several examples including centrifuge modeling tests, finite difference numerical analysis and measured data in situ. Furthermore, the parametric analysis to existing pipelines in several representative layered soils in Shanghai is also carried out. The results discussed in this paper indicate that the Galerkin solution can estimate the pipeline mechanical behavior affected by tunneling in homogeneous soil with good precision. The layered transfer matrix solution is more suitable to simulate the soil stratifications on the pipeline behavior than the Galerkin solution. & 2013 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved. Keywords: Tunnel; Pipeline; Interactional mechanics; Simplified analysis; Galerkin solution; Layered transfer matrix solution

1. Introduction

n Corresponding author at: School of Environment and Architecture, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China. E-mail address: [email protected] (Z. Zhang). Peer review under responsibility of The Japanese Geotechnical Society.

In recent years, the rapid growth in urban development has resulted in an increased demand to develop underground transportation systems. Shield tunneling has become more and more widely used in subway construction in soft soils to reduce interference with surface traffic. However, the tunneling process will inevitably cause inward soil movements around the opening due to the stresses released by tunneling. If these movements become excessive, they may cause serious damage to adjacent existing structures (e.g., buildings, metro tunnels, piles, and pipelines). Boscardin and Cording (1989), Loganathan et al. (2001), Franzius

0038-0806 & 2013 The Japanese Geotechnical Society. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sandf.2013.06.007

558

Z. Zhang, M. Zhang / Soils and Foundations 53 (2013) 557–568

et al. (2004), Jacobsz et al. (2004), Sung et al. (2006) and Shahin et al. (2011) investigated the soil–structure interaction effects induced by tunneling. Many documented case histories indicate that excessive deformation may induce a crack in tunnel segments and eventually may affect the safety and normal use of metro trains (Cooper et al., 2002; Clayton et al., 2006). Therefore, one of the important issues of shield tunneling in urban areas is the estimation of mechanical behavior of adjacent existing pipelines induced by undercrossing tunneling. Recently, some attempts have been made to research the response analysis of existing tunnels due to adjacent tunneling. Methods for solving the problem may be classified into three categories: physical model tests, numerical simulated methods, and simplified analytical methods. Physical model tests, such as, centrifuge modeling, have served an important role in investigating the interaction mechanisms between existing pipelines and newly built tunnels (Kim et al., 1998; Vorster et al., 2005a; Byun et al., 2006; Marshall et al., 2010a). Another major method used to solve the problem is the numerical simulated analysis (Soliman et al., 1998; Addenbrooke and Potts, 2001; Chehade and Shahrour, 2008). The numerical simulated method can take full account of the complex excavation sequence and the soil elastoplastic characters. The commercial software is generally needed in order to form the complex element discrete model. In addition, a simplified analytical method to analyze such a problem may be carried out in two steps: first, the estimation of green-field ground movements induced by tunneling, which would occur if the existing pipelines were not present; second, the calculation of the response of the existing pipelines to green-field ground movements. The conventional simplified analytical approach to solve this problem utilizes the Winkler model such as that proposed by Attewell et al. (1986). The model allows a convenient description of nonlinear soil–tunnel interaction through a single degree of freedom load–displacement relation (i.e., spring coefficient). Vesic (1961) equation is usually expressed to the soil subgrade modulus. Considering that the Winkler model is localized and takes no account of the continuous quality of the foundation deformation, a more rigorous elastic continuum solution is presented by Klar et al. (2005) and Vorster et al. (2005b) based on the homogeneous half space model. Klar et al. (2007) extended the elastic continuum solution to include local yielding around the pipeline, and Klar et al. (2008) estimated the behavior of jointed pipelines in the continuum elastic formulation. All the above solutions are based on the assumption that the foundation may be represented as a homogeneous, isotropic, elastic half space system, which is not consistent with the actual situation of the subsoil. For most of the geotechnical situations, however, layered formations with different material properties are usually encountered in situ. Therefore, it is essential to consider the soil stratified characters in order to fully simulate the deformation behavior of the practical foundation. Classical studies on this topic for the layered medium can be found in Burmister (1945), who developed an elasticity theory for axisymmetric contacts and obtained solutions for the two-layered and threelayered soils.

Since these classical studies, the analyses of multi-layered material regions subjected to axisymmetric loads have been extensively carried out with the method of Hankel transformation and transfer matrix technique in the cylindrical coordinate (Wang and Ishikawa, 2001; Lu and Hanyga, 2005; Han, 2006; Pan et al., 2007). It is convenient to adopt the cylindrical coordinate to solve axisymmetric problems, since the basic equations can be easily converted into the state equations by Hankel transformation. In addition, the several theoretical studies have conducted to overcome asymmetric problems in the cylindrical coordinate (Davies and Banerjee, 1978; Ai et al., 2002; Fukahata and Matsu'ura, 2005). In their methods, the field variables and asymmetric loads are expressed in terms of the Fourier series expansion. However, their methods have some disadvantages, including complicated derivation process and formula, slow convergence and even not convergence of the trigonometric series. Most of the aforementioned research is focus on the solutions subjected to external loads located on the surface ground. Little attention has been paid to considering the condition with internal loads in the layered medium (Davies and Banerjee, 1978; Ai et al., 2002; Fukahata and Matsu'ura, 2005; Lu and Hanyga, 2005). Since they take into consideration the many asymmetric problems encountered in situ, as well as the loads located arbitrarily in practical projects, the current methods based on the cylindrical coordinate system are rather complicated. According to this study, the proposed solution in the Cartesian coordinate is the preferred approach to solve problems involving internal loads in multi-layered soil. Based on the above-mentioned layered soil foundational solution, a layered transfer matrix solution is presented to analyze the mechanical behavior of adjacent pipelines induced by tunneling. In order to compare with the effects of non-homogeneous soil characteristics on the structural deformations, a Galerkin method is also proposed here. Actually, this current study is indeed a decoupling analysis: firstly, estimating green-field soil movements induced by tunneling; secondly, calculating the pipeline response to these soil movements. Zhang et al. (2012) presented a coupling numerical method to reflect the coupling effects of tunnel–soil– pipeline interaction by combined finite element and boundary element approach. The main aim of current study is to pursue for a simplified decoupling method and conduct a meaningful comparison between Galerkin method and layered transfer matrix solution. Their basic assumptions and input parameters must be the same, and the assumptions are as follows: (1) the pipeline is continuous and always in contact with the surrounding soils; (2) the tunnel is unaffected by the existence of the pipelines; (3) the pipeline is regarded as Euler–Bernoulli beam. The Hermite element of two nodes and four degrees of freedom are utilized in their studies. (4) the green-field soil movements are represented by the analytical solution proposed by Loganathan and Poulos (1998). Specifically, assumption (2) simply means that the tunnel exhibits the same behavior as it would if there was no pipeline. This is an essential assumption in this study, allowing for the decoupling of tunnel behavior in the solution of the pipeline response through the use of a green-field settlement trough. In addition, the closed-form solution for tunneling-induced soil movements is one of the focal points for many geotechnical engineers. Verruijt and Booker

Z. Zhang, M. Zhang / Soils and Foundations 53 (2013) 557–568

(1996) presented an approximate solution for the problem in a homogeneous elastic half space, by extending the method suggested by Sagaseta (1987) for the case of ground loss in an incompressible soil. Since they considered the uniform radial ground movement around the tunnel for the short-term undrained condition (Fig. 1(a)), the predicted settlement troughs are wider and horizontal movements are larger than observed values. In order to consider the actual oval-shaped ground deformation pattern (Fig. 1(b)), Loganathan and Poulos (1998) presented a modified solution from Verruijt and Booker (1996) by suggesting the use of an equivalent ground loss ratio, which can be estimated using the gap parameter proposed by Lee et al. (1992). Therefore, the solutions proposed by Loganathan and Poulos (1998) are adopted to calculate the green-field settlements in this study. 2. Galerkin solution Fig. 2 shows a schematic diagram of this study, in which a new tunnel is excavated under an existing pipeline. The deformation behavior of the pipeline subjected to the soil movements induced by tunneling can be analyzed by assuming the soil to be modeled by the modulus of subgrade reaction. The soil pressure p acting on the pipeline can be expressed p ¼ k½wp ðxÞ−w0 ðxÞ

ð1Þ

Excavation opening

where k is defined as the subgrade coefficient. Attewell et al. (1986) suggested the use of the Vesic (1961) equation for the subgrade modulus, which is given by sffiffiffiffiffiffiffiffiffiffi 0:65E s 12 E s D4 ð2Þ k¼ 1−μ2 EI in which D is the outer diameter of pipeline, EI is the bending stiffness of pipeline. E s is the elastic modulus of soil, μ is Poisson's ratio of soil. According to non-homogeneous foundation, the soil elastic parameters under the condition of homogeneous foundation are calculated by the means of weighted average proposed by Poulos and Davis (1980). wp ðxÞ is the vertical displacement of pipeline; w0 ðxÞ is the green-field vertical displacement due to tunneling, which can be calculated by the solutions proposed by Loganathan and Poulos (1998), that is  z0 −h z0 þ h w0 ðxÞ ¼ ε0 R2 − þ ð3−4μÞ 2 2 2 x þ ðz0 −hÞ x þ ðz0 þ hÞ2  2z0 ½x2 −ðz0 þ hÞ2  −f½1:38x2 =ðhþRÞ2 þ½0:69z0 2 =h2 g − ð3Þ ⋅e ½x2 þ ðz0 þ hÞ2 2 in which R and h are the radius and embedment depth of the tunnel, z0 is the embedment depth of the pipeline. ε0 is the ground loss ratio proposed by Lee et al. (1992). The governing differential equation for the tunnel–soil– pipeline interaction is given by EI

Tunnel

Fig. 1. Deformation patterns around the tunnel section: (a) uniform radial; (b) oval-shaped.

Surface ground

o x

Layer one

Layer k

Existing pipeline

Tunnel

Layer m

Layer n Half space

z

Fig. 2. Schematic representation for tunnel–soil–pipeline interaction.

559

d 4 wp ðxÞ þ Kwp ðxÞ ¼ Kw0 ðxÞ dx4

ð4Þ

in which K is the subgrade coefficient per unit length of the pipeline, and K ¼ kD. Eq. (4) is fourth-order non-homogeneous differential equation. The solution can be obtained using the finite element approach in which the pipeline is represented by Euler– Bernoulli beam elements based on assumption 3. The displacement variable wp ðxÞ is approximated in terms of discrete nodal values as follows: wp ðxÞ ¼ N 1 wi þ N 2 θi þ N 3 wj þ N 4 θj

ð5Þ

where wi and θi are the vertical and rotational displacement at node i, respectively; wj and θj are the vertical and rotational displacement at node j, respectively. The shape functions N i ði ¼ 1; 2; 3; 4Þ are defined as follows: N 1 ¼ ðl3 −3lx2 þ 2x3 Þ=l3

ð6aÞ

N 2 ¼ ðl2 x−2lx2 þ x3 Þ=l2

ð6bÞ

N 3 ¼ ð3lx2 −2x3 Þ=l3

ð6cÞ

N 4 ¼ ðx3 −lx2 Þ=l2

ð6dÞ

where l is the unit length of the beam element. Eq. (5) can be written in matrix form as wp ðxÞ ¼ fNgfwp ge

ð7Þ

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Z. Zhang, M. Zhang / Soils and Foundations 53 (2013) 557–568

where fNg is the interpolation function matrix, fNg ¼  N 1 N 2 N 3 N 4 gT .fwp ge is the displacement vector of h iT the beam element e, fwp ge ¼ vi θi vj θj . Based on the shape functions, the form of element matrices for the soil and pipeline can be expressed as follows: Z l e KfNgfNgT dx ð8Þ ½K s  ¼ 0

Z

l

½K p  ¼ e

o



d2 N EI dx2



d2 N dx2

T dx

ð9Þ

Applying the Galerkin method to the governing differential equation in Eq. (4) yields the following elements matrix form: Z l e e e e KfNgw0 ðxÞ dx ð10Þ ½K s  fwp g þ ½K p  fwp g ¼ 0

ð11Þ

where fPg is the element force vector acting on the beam due to tunneling, that is, Z l e KfNgw0 ðxÞ dx ð12Þ fPg ¼ e

0

The longitudinal deformation displacement fwp g for existing pipeline may be represented by the following relation after the assembly of element matrices: ð½K s  þ ½K p Þfwp g ¼ fPg

ð13Þ

in which ½K s  is the global stiffness matrix of soil, ½K p  is global stiffness matrix of pipeline, fPg is the global matrix of force vector acting the beam due to tunneling. For a given set of soil movements induced by tunneling in Eq. (3), the deformations of the pipeline can be determined by solving Eq. (13), and the bending moments obtained from the resulting pipeline deformations M p ðxÞ ¼ −EI

d2 wp ðxÞ dx2

n

wsi ¼ ∑ ζ ij f sj

ð16Þ

j¼1

where wsi is the soil displacement at the arbitrary point i, soil flexibility coefficient ζ ij is the soil displacement at point i due to the unit load at point j, f sj is the force acting on the point j of the soil medium. This soil displacement can be decomposed into two components: ws−own is the displacement at the point due to its own i loading, and wis−other is the additional displacement of the point due to loading at different locations (i.e., at the points along the pipeline or beside the tunneling excavation): n

wsi ¼ wis−own þ wis−other ¼ ζ ii f si þ ∑ ζ ij f sj

ð14Þ

j≠i

The additional displacement wis−other in Eq. (17) can be other further decomposed into two parts: ws−interaction is the i additional displacement caused by interaction forces at other locations along the pipeline (at other locations than i), and ws−tunnel is the additional displacement due to the tunneling: i wsi ¼ wis−own þ wis−interaction other þ ws−tunnel i ¼ ζ ii f si þ

n

∑

j¼1 j≠i; j≠tunnel

ζ ij f sj þ w0i

ð18Þ

where w0i is the green-field displacements due to tunneling. By utilizing assumption 4, it can be calculated by the solutions proposed by Loganathan and Poulos (1998). Considering that the forces acting on the pipeline are equal but opposite to the forces acting on the soil F pi ¼ −f si ¼ −

ws−own i ζ ii

ð19Þ

Due to displacement compatibility relation, the displacements of pipeline are equal to those of the soil medium other wsi ¼ wli ¼ wis−own þ ws−interaction þ wis−tunnel i

ð20Þ

Introducing Eqs. (18)–(20) into Eq. (15) results in

3. Layered transfer matrix solution

ð½K p  þ ½K sl Þfwp g ¼ ½K sl fws−interaction other g þ ½K sl fws−tunnel g ð21Þ

3.1. Mechanical analysis for tunnel–soil–pipeline interaction The Euler–Bernoulli beam is applied to calculate the bending problem in this study based on assumption 3. The same interpolation functions in Eqs. (6a)–(6d) are utilized to simulate these Hermite elements. The deformation behavior of existing pipelines can be represented by the following relation: ½K p fwp g ¼ fF p g

ð17Þ

j¼1

where ½K s e and ½K p e are the soil element matrix and pipeline element matrix for the unit e and they are calculated by Eqs. (8) and (9). Eq. (10) can be expressed in the following form: ð½K s e þ ½K p e Þfwp ge ¼ fPge

The soil displacement can be evaluated using soil flexibility coefficients:

ð15Þ

where ½K p  is the global stiffness matrix of pipeline, fwp g is the global displacement vector, fF p g is the global force vector representing soil loads acting on the beam elements.

in which ½K sl  is a diagonal matrix, the element K ij ¼ 1=ζ ii for i ¼ j and 0 for i≠j. It should be noted that fws−interaction other g ¼ ½ζff s g−½ζ 0 ff s g ¼ −ð½ζ−½ζ 0 Þ½K p fwp g ð22Þ where ½ζ is the soil flexibility matrix, the definition for element ζ ij is same with Eq. (16). ½ζ 0  is a diagonal matrix, the element ζ 0ij ¼ ζ ij for i ¼ j and 0 for i≠j. By introducing Eq. (22) to Eq. (21) and rearranging the terms, the deformation behavior of existing pipeline affected

Z. Zhang, M. Zhang / Soils and Foundations 53 (2013) 557–568

by adjacent tunneling can be obtained: ðI þ ½ζ½K p Þfwp g ¼ fw0 g

ð23Þ

where I is a unit matrix. fw0 g is the green-field displacement vector. It should be noted that the element ζ ij for the matrix ½ζ are defined as the soil vertical displacement at node i due to the unit load at node j. They can be evaluated by a fundamental solution for the vertical displacement of a point within an elastic, stratified medium caused by the vertical point load within the medium, which will be introduced below. 3.2. Fundamental solution for layered soil model In this study, the fundamental solution for multi-layered soils subjected to a vertical point load in the Cartesian coordinate will be applied to construct the components of ½ζ in Eq. (23). The layered soil model shown as Fig. 2 consists of n (n≥1) parallel, elastic isotropic layers lying on a homogeneous elastic half space. Each layer has own Young's modulus E i and Poisson's ratio μi . The ith layer occupies a layer region hi−1 ≤ z ≤ hi of thickness Δhi (Δhi ¼ hi −hi−1 ), where i ¼ 1; 2; …; or n. The vertical point load P is assumed to set at a point ðx0 ; y0 ; hm1 Þ in the mth layer. The double Laplace integral transform and inverse double Laplace transform will be applied to transfer between the transform domain and physics domain: Z ∞Z ∞ ~ η; zÞ ¼ φðξ; φðx; y; zÞe−ðξxþηyÞ dx dy ð24Þ 0

1 φðx; y; zÞ ¼ − 2 4π

0

Z

βþi∞ β−i∞

Z

βþi∞ β−i∞

~ η; zÞe φðξ;

ξxþηy

dξ dη

− uðx; y; hþ i Þ ¼ uðx; y; hi−1 Þ

ð28aÞ

− vðx; y; hþ i Þ ¼ vðx; y; hi−1 Þ

ð28bÞ

− wðx; y; hþ i Þ ¼ wðx; y; hi−1 Þ

ð28cÞ

− τzx ðx; y; hþ i Þ ¼ τzx ðx; y; hi−1 Þ

ð28dÞ

− τzy ðx; y; hþ i Þ ¼ τzy ðx; y; hi−1 Þ

ð28eÞ

− sz ðx; y; hþ i Þ ¼ sz ðx; y; hi−1 Þ−qðx; y; hm1 Þ − sz ðx; y; hþ i Þ ¼ sz ðx; y; hi−1 Þ

ðfor z ¼ hm1 Þ

ðfor z≠hm1 Þ

ð28fÞ ð28hÞ

where hi is the distance from the bottom of the ith layer to the surface of the first layer (i ¼ 2; 3; …; or n); the superscripts “+” and “−” denote the values of the functions just on upper and lower interface boundary of the ith layer; qðx; y; hm1 Þ denotes the surface density distribution of the point load Pðx0 ; y0 ; hm1 Þ, that is, qðx; y; hm1 Þ ¼ Pðx0 ; y0 ; hm1 Þδðx−x0 ; y−y0 Þ

ð29Þ

in which δðx−x0 ; y−y0 Þ is the Dirac singularity function. The state variable vector for displacements and stresses at two boundary surfaces z ¼ 0 and z ¼ hn can be expressed in the transform domain

ð25Þ

τzx ðx; y; 0Þ ¼ τzy ðx; y; 0Þ ¼ sz ðx; y; 0Þ ¼ 0

ð26Þ

uðx; y; hn Þ ¼ vðx; y; hn Þ ¼ wðx; y; hn Þ ¼ 0

ð27Þ

2 6 6 6 6 0 6 6 6 μ 6 ξ 6 6 μ−1 Θðξ; ηÞ ¼ 6 6 E 2 E 2 6 6 μ2 −1 ξ − 2ð1 þ μÞ η 6 6 E 6 ξη 6 2ðμ−1Þ 4 0

Assuming that the stresses and displacements located at the each interface between two connected layers are completely continuous, and the load surface is considered as an artificial interface (z ¼ hm1 ), it can be expressed

~ η; 0Þ ¼ ½uðξ; ~ η; 0Þ v~ ðξ; η; 0Þ wðξ; ~ η; 0Þ τ~ zx ðξ; η; 0Þ τ~ zy ðξ; η; 0Þ s~ z ðξ; η; 0ÞT Gðξ;

where ξ and η are the integration parameters for the Laplace transform. According to a traction free condition at the ground surface of the layered system, and a fixed boundary condition at the bottom (hn approaches ∞), it can be obtained

0

561

ð30Þ ~ η; h− Þ ¼ ½uðξ; ~ η; h−n Þ v~ ðξ; η; h−n Þ wðξ; ~ η; h−n Þ Gðξ; n τ~ zx ðξ; η; h−n Þ τ~ zy ðξ; η; h−n Þ s~ z ðξ; η; h−n ÞT

Based on the transfer matrix method (Ai et al., 2002; Pan et al., 2007), the transfer function Φðξ; η; zÞ is defined in current study, that is Φðξ; η; zÞ ¼ exp½zΘðξ; ηÞ

ð32Þ

where

3

0

−ξ

2ð1 þ μÞ E

0

−η

0

2ð1 þ μÞ E

0

0

0

0

0

0

0

0

0

0

−ξ

−η

μ η μ−1 E ξη 2ðμ−1Þ E 2 E ξ2 η − 2 μ −1 2ð1 þ μÞ 0

ð31Þ

0

0

7 7 7 7 0 7 7 27 1−μ−2μ 7 7 Eð1−μÞ 7 7 7 μ ξ 7 7 μ−1 7 7 μ η 7 7 μ−1 5 0

ð33Þ

562

Z. Zhang, M. Zhang / Soils and Foundations 53 (2013) 557–568

By introducing the double Laplace integral transform in Eq. (24) to continuity conditions in Eqs. (28a)–(28h), and using the transfer function in Eq. (32), the equation governing the relations for Eqs. (30) and (31) can be obtained as ~ η; h− Þ ¼ ½F 1 Gðξ; ~ η; 0Þ−½F 2 fQg ~ Gðξ; ð34Þ n

~ is the load vector in the transform domain, that is, where fQg ~ ¼ ½ 0 0 0 0 0 q~ðξ; η; hm1 Þ T ð35Þ fQg where ½F 1  and ½F 2  are the global transfer matrices, that is, ½F 1  ¼ Φðξ; η; Δhn ÞΦðξ; η; Δhn−1 Þ⋯Φðξ; η; Δh1 Þ

ð36Þ

½F 2  ¼ Φðξ; η; Δhn ÞΦðξ; η; Δhn−1 Þ⋯Φðξ; η; Δhm2 Þ

ð37Þ

where Δhi is the thickness of the ith layer with Δh1 ¼ h1; Δhi ¼ hi −hi−1 (i ¼ 2; 3; …; or n), and Δhm2 ¼ hm −hm1 . Using the two boundary conditions of Eqs. (26) and (27), ~ η; 0Þ and Gðξ; ~ η; h− Þ in Eq. (34) can be determined the Gðξ; n analytically. The stresses and displacements in the transform domain at depth z in the ith layer above or below the artificial interface (z ≤ hm1 or z4 hm1 ) can be expressed as follows: ~ η; zÞ ¼ ½λ1 Gðξ; ~ η; 0Þ ðfor z ≤hm1 Þ Gðξ; ð38Þ ~ η; zÞ ¼ ½λ2 Gðξ; ~ η; h− Þ Gðξ; n

ðfor z4 hm1 Þ

relative density of 90% using an automatic sand pourer. The tunnel had an outer diameter of 62 mm and was buried at a depth of 182 mm. A pipeline was placed at a depth of 70 mm and had an outer diameter of 19.06 mm and a wall thickness of 1.63 mm. In model scale, the pipeline has a bending stiffness of 238.5 N m2. The ground losses applied in the tests controlled by a motor-driven actuator were 0.3%, 1% and 2.5%, corresponding to the lower, upper, and higher ranges of typical soil loss. The schematic representation for centrifuge model test is shown as Fig. 3. The layered transfer matrix solution can also be applied to homogeneous soil by dividing the whole soil into multiple Strong-box

Soil surface 70

19.06 Model pipeline

182

312

Model tunnel

ð39Þ

62 Soil body

where ð40Þ

½λ2  ¼ Φðξ; η; z−hi ÞΦðξ; η; −Δhiþ1 Þ⋯Φðξ; η; −Δhn Þ

ð41Þ

Introducing the inverse double Laplace transform of Eq. (25) ~ η; zÞ in Eqs. (38) and (39), the solution for into the solution Gðξ; stresses and displacements in the layered soils subjected to the vertical load can be obtained in the physics domain. When Pðx0 ; y0 ; hm1 Þ ¼ 1, the solution is the fundamental solution for layered soils subjected to a vertical unit point load. 4. Example validation and parametric analysis

770

Fig. 3. Schematic representation for centrifuge model test.

Vertical displacement (mm)

½λ1  ¼ Φðξ; η; z−hi−1 ÞΦðξ; η; Δhi−1 Þ⋯Φðξ; η; Δh1 Þ

By the approach discussed above, computer programs for the Galerkin solution and the layered transfer matrix solution have been written for estimating the existing pipeline behavior induced by tunneling.

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -300

Observation (ε=0.3%) Layered transfer matrix solution (ε=0.3%) Galerkin solution (ε=0.3%)

-200

-100 0 100 200 Offset from tunnel centerline (mm)

300

Fig. 4. Comparisons of pipeline vertical displacement (ε¼ 0.3%).

Vertical displacement (mm)

4.1. Example validation 4.1.1. Tunnel in homogeneous soil (comparison with centrifuge model tests) Marshall et al. (2010b) and Marshall and Mair (2008) carried out a series of centrifuge model tests to observe the effects of tunneling on adjacent pipelines. All tests were conducted at an acceleration level of 75 g. The centrifuge strong-box had plan dimensions of 770  147.5 mm and was filled with Leighton Buzzard Fraction E silica sand to a depth of 312 mm. The sand had a typical average grain size of 122 μm, a specific gravity of 2.67, maximum and minimum void ratios of 0.97 and 0.64, respectively, and was poured to a

147.5

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -300

Observation(ε=1%) Layered transfer matrix solution (ε=1%) Galerkin solution (ε=1%)

-200

-100 0 100 200 Offset from tunnel centerline (mm)

Fig. 5. Comparisons of pipeline vertical displacement (ε ¼1%).

300

Z. Zhang, M. Zhang / Soils and Foundations 53 (2013) 557–568

Vertical displacement (mm)

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -300

Observation(ε=2.5%) Layered transfer matrix solution (ε=2.5%) Galerkin solution (ε=2.5%)

-200

-100 0 100 200 Offset from tunnel centerline (mm)

300

Fig. 6. Comparisons of pipeline vertical displacement (ε¼ 2.5%).

Bending moment (N.m)

2 1 0 -1 -2 -3 -4

Observation (ε=0.3%) Layered transfer matrix solution (ε=0.3%) Galerkin solution (ε=0.3%)

-5 -6 -300

-200

-100 0 100 200 Offset from tunnel centerline (mm)

300

Fig. 7. Comparisons of pipeline bending moment (ε ¼0.3%).

Bending moment (N.m)

2 1 0 -1 -2 -3 -4

Observation (ε=1%) Layered transfer matrix solution (ε=1%) Galerkin solution (ε=1%)

-5 -6 -300

-200

-100 0 100 200 Offset from tunnel centerline (mm)

Fig. 8. Comparisons of pipeline bending moment (ε¼ 1%).

300

2 Bending moment (N.m)

layers with equal elastic characteristics. A comparison with the Galerkin solution for homogeneous soil is also presented here. Figs. 4–6 show the pipeline vertical displacements measured in the centrifuge tests and those predicted by the layered transfer matrix solution and Galerkin solution. The comparisons of pipeline bending moments calculated by the proposed solutions and those observed are shown in Figs. 7–9. From the above figures, it can be seen that the calculated displacement and bending moment for pipelines using layered transfer matrix solution are in general consistent with the results using the Galerkin solution. It shows that good agreement is obtained between the two proposed methods when applied to homogeneous soil. In addition, the comparisons show that in the

563

1 0 -1 -2 -3 -4

Observation (ε=2.5%) Layered transfer matrix solution (ε=2.5%)

-5

Galerkin solution (ε=2.5%)

-6 -300

-200

-100 0 100 200 Offset from tunnel centerline (mm)

300

Fig. 9. Comparisons of pipeline bending moment (ε¼ 2.5%).

case of soil losses of 0.3% and 1%, the calculated curves, including the Galerkin solution and layered transfer matrix solution, compare well with the observed one. The discrepancy between the calculated and the measured data increases with increasing soil losses of 2.5%. Several factors, including the non-elastic soil behavior, the behavior of the soil–pipe interaction, and the stiffness degradation of the soil, may be the reasons for this larger deviation. With increasing tunnel losses, tunneling-induced soil movement will degrade the soil stiffness due to the corresponding shear strain, and the soil elastic behavior will be treated as either nonlinear or elastic–plastic. It should be also noted that the large relative displacement between the soil and pipeline may induce slippage at the interface. The different slippage behavior between the soil and pipeline can be affected by different surface smooth degrees and pipeline–soil material stiffnesses. According to typical representatives of polyethylene, concrete, and steel pipelines, in general, steel and concrete pipelines may be well represented using this current method. For polyethylene pipelines predictions using this current method may be deviate significantly from elastic predictions. In addition, if the condition is for the bigger soil loss, the proposed method may underestimate the deformation behavior for existing pipelines, so the proposed method should be used with caution. All the factors discussed above are beyond the scope of this paper, which focuses on elastic solutions. However, several factors such as relative slippage failure and gapping between the existing pipelines and the surrounding soil, which would contribute additionally to nonlinear soil behavior, should be introduced into the analysis in near future. 4.1.2. Tunnel in non-homogeneous soil (comparison with 3D numerical simulation analysis) One tunneling case is selected to demonstrate the effects of soil non-homogeneity on the deformation behavior of existing pipelines. A sewer pipeline 3.5 m in outer diameter and 0.33 m thick exists perpendicular to and above the tunnel, buried 9.37 m below the ground surface. Its bending stiffness is 7.23  107 kN m2. Excavation of the tunnel (17.05 m in embedment depth, 6.2 m in outer diameter, and 5.5 m in inner diameter) is carried out by earth pressure balance shield machine with an outside diameter of 6.34 m. The six-layered soil properties are listed in Table 1. In order to compare with the Galerkin solution and layered transfer matrix solution, a mixed analytical–numerical approach

Z. Zhang, M. Zhang / Soils and Foundations 53 (2013) 557–568 -5

Table 1 Geotechnical characteristics for six-layered soils. Layer number

Thickness (m)

Elastic modulus (MPa)

Poisson's ratio

① ② ③ ④ ⑤ ⑥

1.75 1.15 10.3 2.85 3.35 6.35

8.86 21.18 29.75 7.98 9.73 13.93

0.33 0.25 0.24 0.35 0.32 0.26

Vertical displacement (mm)

564

Numerical results layered transfer matrix solution

0

Galerkin solution

5 10 15 20 -50

-40

-30

-20 -10 0 10 20 30 Offset from tunnel centerline (m)

40

50

40

50

Fig. 11. Comparisons of pipeline vertical displacement.

Bending moment (kN.m)

-500 0 500 1000 1500 Numerical results layered transfer matrix solution Galerkin solution

2000 2500

Fig. 10. 3D element mesh for soil–pipeline interaction.

is used based on the large-scale commercial software. The finite difference code FLAC3D is employed to solve the soil–pipeline interaction, based on the closed form green-field displacements in Eq. (3). Fig. 10 shows the 3D mesh used in the analysis. The dimension is taken as 100 m, 40 m, and 30 m along the x; y; z coordinate direction. The finite difference code was not used to simulate and generate the tunneling process, but to directly evaluate the mechanical behavior for pipelines based on the closed form green-field soil displacements in Eq. (3). This is an essential item in this current work. Otherwise, the input would not have been the same and the comparisons with the current solution would be meaningless. The identical solving approach is very important to their comparisons. The Galerkin method and layered transfer matrix method are indeed two-steps analysis, firstly estimation of green-field ground movements induced by tunneling and secondly calculation of the response of existing pipelines to green-field ground movements. The mixed analytical–numerical approach is same with the twosteps analysis. At the first step for the FLAC3D, the bending stiffness of pipeline is set to zero and the element mesh is forced to deform according to Eq. (3) which is simulated to develop the green-field displacements due to tunneling. Then the unbalanced nodal forces are extracted from FLAC3D and stored in memory. At the second step, the pipeline elements are given its actual stiffness, and the stored forces from the first step are applied to the nodes. The model boundaries belong to the displacement controlled type which is forced to move according to green-field displacements. The comparisons of vertical displacements and bending moments for pipelines by the proposed methods and numerical results are shown in Figs. 11 and 12. As for the subgrade

-50

-40

-30 -20 -10 0 10 20 30 Offset from tunnel centerline (m)

Fig. 12. Comparisons of pipeline bending moment. Table 2 Geotechnical characteristics in situ. Layer number

Thickness (m)

Elastic modulus (MPa)

Poisson's ratio

① ② ③ ④

16.6 1.82 3.98 3.95

8.2 25 52.9 150

0.3 0.2 0.21 0.2

modulus in Galerkin method, the elastic parameters of soils under the condition of homogeneous foundation are calculated by the means of weighted average proposed by Poulos and Davis (1980). These figures show that the solutions from the layered transfer matrix method are in good agreement with those from the FLAC3D analysis. Generally speaking, the assumption that the tunnel is unaffected by the existence of the pipelines is also a key point in the above calculations. It essentially means that the tunnel–soil–pipeline interaction is composed of a superposition of green-field displacements due to tunneling and soil–pipeline interaction. From the above comparisons, it is observed that the poor agreement between the solutions from the Galerkin method and those from the numerical software is obtained and the proposed Galerkin method underestimates the pipeline's deformation. It appears that the layered transfer matrix method is a valid approach to estimate the mechanical deformation for existing pipeline induced by tunneling in non-homogeneous soils and the soil non-homogeneity has significant effects on pipeline deformation. Furthermore, the error obtained by the Galerkin method

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based on weighted average cannot be dismissed when dealing with the layered soils where the difference of elastic parameters among successive layers is large.

Table 3 Geotechnical characteristics for overlying hard two-layered soils. Layer number

Thickness (m)

Elastic modulus (MPa)

Poisson's ratio

4.1.3. Tunnel in non-homogeneous soil (comparison with measured data in situ) The tunnel from Yitian Station to Xiangmihu Station is an important part of Shenzhen railway transportation line, which is carried out by a shield machine with an outside diameter of 6.19 m. The outer diameter of tunnel segments is 6 m and the tunnel depth is 14.5 m. There is a cable pipeline (8.7 m in embedment depth, 3 m in outer diameter and 12 cm thickness) perpendicular to and above the tunnel. Its bending stiffness is 2.82  107 kN m2. Soil properties from the reported ground investigation are listed in Table 2. According to the tunnel monitoring scheme in situ (Ma, 2005; Jia et al., 2009), two separate series of points are marked on the east and west inner walls of the pipeline to measure the pipeline deformation. A comparison of the calculated and observed pipeline displacements is shown in Fig. 13. As for the Galerkin solution, the elastic parameters of homogeneous soil are calculated by the means of weighted average proposed by Poulos and Davis (1980). It is clear that the predictions from the layered transfer matrix solution are in general consistent with the observed data. The calculated sagging of the pipeline displacement is deeper than measured results and that the calculated maximum displacement is larger, which offers a conservative estimate of the pipeline deformation induced by tunneling. In addition, the figure also shows that the Galerkin solution provides smaller vertical displacement for the pipelines and underestimates the pipeline deformation.

① ②

20 20

20 5

0.35 0.35

4.2. Parametric analysis A variety of complex strata with different soil material properties are usually encountered in China's coastal regions. For example with Shanghai, the typical stratigraphic distribution can be summarized as: the first layer is the brown clay (i.e., the hard surface); the second layer is the loamy silty clay or clayey silt, the third layer is the gray silty clay, sap green silty clay or grass yellow sandy silt. Previous studies about such routine parameters as tunnel-induced soil movement and the soil–pipeline interaction in homogeneous soil have been

Pipeline settlement (mm)

-2

Measured in situ(East point) Measured in situ(West point) Galerkin solution Layered transfer matrix solution

0 2 4 6 8 10 -50

-40

-30

-20 -10 0 10 20 30 Offset from tunnel centerline (m)

40

Fig. 13. Comparisons between calculated and measured values.

50

Table 4 Geotechnical characteristics for underlying hard two-layered soils. Layer number

Thickness (m)

Elastic modulus (MPa)

Poisson's ratio

① ②

20 20

5 20

0.35 0.35

Vertical displacement (mm)

-2 0 2 4 6 8

Layered transfer matrix solution(Overlying hard) Layered transfer matrix solution(Underlying hard)

10 -15

-10

-5 0 5 Offset from tunnel centerline (m)

10

15

Fig. 14. Comparisons of pipeline vertical displacement.

published. Therefore, this study attempts to investigate only the influence of soil stratification on the pipeline's behavior due to tunneling. 4.2.1. Two-layered soils Assume that the outer diameter of an existing pipeline is 0.4 m, the bending moment, 1.5  104 kN m2, the axis depth, 20 m. The outer diameter of a tunnel is 2 m, the axis depth, 26 m. The ground loss ratio is set as 6%. The geotechnical characteristics of overlying hard and underlying hard layered soils are summarized in Tables 3 and 4, respectively. All the comparisons are given below correspond to a Poisson’s ration of 0.35 and a thickness of 20 m for each layer. The soil elastic modulus ratios are set as 4:1 and 1:4 for overlying hard and underlying hard layered soils. This study attempts to investigate the influence of a weak or strong layer on the deformation behavior of pipelines. Fig. 14 shows the comparisons of pipeline vertical displacements by means of the layered transfer matrix solution according to the overlying hard and underlying hard layered soils. It is observed that the overlying hard layered soils provide larger pipeline settlements than the underlying hard layered soils. The effect of preventing pipeline settlements in the underlying hard layered soils is superior to that in the overlying hard layered soils. Figs. 15 and 16 show maximum pipeline vertical displacement with different elastic moduli for the surface and underlying layer,

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Maximum settlement (mm)

14

Table 5 Geotechnical characteristics for three-layered soils (Case 1).

12 10 8

Layer number

Thickness (m)

Elastic modulus (MPa)

Poisson's ratio

① ② ③

1.5 3.5 10

5 10 15

0.35 0.30 0.25

6 4 5

10 15 20 25 Overlying layer's elastic modulus (MPa)

30

Fig. 15. Effects of overlying hard layer's elastic modulus on pipeline vertical displacement.

Table 6 Geotechnical characteristics for three-layered soils (Case 2). Layer number

Thickness (m)

Elastic modulus (MPa)

Poisson's ratio

① ② ③

1.5 3.5 10

6 18 30

0.35 0.30 0.25

12

-2

10 8 6 4 5

10 15 20 25 Underlying layer's elastic modulus (MPa)

30

Fig. 16. Effects of underlying hard layer's elastic modulus on pipeline vertical displacement.

Vertical displacement (mm)

Maximum settlement (mm)

14

0 2 4 6 8 10

Numerical results (Case 1) Numerical results (Case 2) Layered transfer matrix solution(Case 1) Layered transfer matrix solution(Case 2)

12 14 16 -15

-10

-5 0 5 Offset from tunnel centerline (m)

10

15

Fig. 17. Comparisons of pipeline vertical displacement.

respectively. From the above two figures, it appears that the maximum pipeline settlement obviously decreases when the layer's elastic modulus increases, whether it be the hard surface layer or the hard underlying layer. The improvement of the soil modulus can enhance the deformation resistance effects for pipelines on the situ of tunneling.

cases occurred due to the parametric analysis of the pipeline behavior in the layered soils when the difference between the elastic parameters among successive layers was large. 5. Conclusion

4.2.2. Three-layered soils Assume that the outer diameter of an existing pipeline is 0.4 m, the bending moment, 1.05  105 kN m2, the axis depth, 1.5 m. The outer diameter of a tunnel is 1.5 m, the axis depth, 5 m. The ground loss ratio is 5%. The geotechnical characteristics of soils in two cases are summarized in Tables 5 and 6, respectively. The elastic modulus ratios of each layer in the two cases are set as 1:2:3 and 1:3:5, respectively. The thickness and Poisson's ratios of each layer were set as 3:7:20 and 7:6:5, respectively, for the two cases. Fig. 17 shows a comparison of the pipeline vertical displacement for two different layered soils between the proposed layered transfer matrix solution and numerical results based on the FLAC3D. As shown in the figure, very good agreement is obtained, and it appears that the proposed layered transfer matrix solution is in complete accordance with the numerical analysis results. Comparisons for pipeline settlements for Cases 1 and 2 are also provided in Fig. 17. It shows that the obvious differences between the two

The mechanical problem of tunneling effects on existing pipelines is solved using a Galerkin solution and a layered transfer matrix solution. A subgrade modulus based on the Vesic (1961) equation, which was employed by Attewell et al. (1986), is applied in the Galerkin solution to simulate soil– pipeline interaction. In order to consider the effects of soil stratification on the pipeline deformation behavior, a layered soil model is adopted in the layered transfer matrix solution by applying the double Laplace transform and transfer matrix method. The layered soil model is built in a Cartesian coordinate system, whereas solutions usually existed in a cylindrical coordinate before. As long as the continuity interface conditions between the two layers are changed, the foundational solution with the arbitrary internal loads, such as a lateral point load, can be easily solved. Then, the layered soil model in this study can be further used for the response analysis of adjacent tunneling on existing surface buildings and pile foundations, and so on.

Z. Zhang, M. Zhang / Soils and Foundations 53 (2013) 557–568

The layered transfer matrix solution is compared with the Galerkin solution with the examples including centrifuge modeling tests, 3D numerical analysis and measured data in situ. The Galerkin solution can estimate the pipeline mechanical behavior affected by tunneling in homogeneous soil with good precision. However, in layered soils in which the differences of elastic parameters among successive layers are large, the Galerkin solution, which is treating the soil as homogeneous, will result in significant error. In addition, the layered transfer matrix solution has proven effective in solving this problem for both homogeneous and non-homogeneous layered soils. Specially by comparing with the more rigorous 3D finite difference analysis, good agreement is observed between the two methods, suggesting the proposed layered transfer matrix solution is capable of adequately taking account of soil stratification. The analysis of pipeline behavior in the typical stratigraphic soils shows that soil non-homogeneity has significant effects on pipeline deformation and should be fully considered in the design and construction to reduce potential excavation risks. It should also be noted that the major limitation of the proposed methods stem from the simplified assumptions of linearity and elasticity. For a given green-field soil settlement trough, any soil nonlinearity or elasto-plasticity, whether resulting from pipe–soil interaction or from global soil shearing due to the tunnel, may reduce the maximum bending moment in the pipeline. Any additional reduction in the soil stiffness may result in an upper approximation of the bending moment, as long as the estimated soil stiffness is higher than the true one. Advanced mechanisms such as relative uplift failure and gap between the existing pipeline and soils, advanced elasto-plastic or elasto-viscoplastic constitutive models for soils, should be introduced into this study. The suggested methods do not consider the effect of pipeline joints allowing rotation or axial movement. Therefore, further research on this subject is still required in order to more effectively evaluate the interaction problem for tunnel–soil–pipeline. Acknowledgments The authors acknowledge the financial support provided by Natural Science Foundation of China for Young Scholars (No. 51008188), and by Open Project Program of State Key Laboratory for Geomechanics and Deep Underground Engineering (No. SKLGDUEK1205), and by China Postdoctoral Science Foundation (No. 201104266), and by Shanghai Science and Technology Talent Plan Fund (No. 11R21413200). The authors wish to express their sincere gratitude to Prof. Huang M S at Tongji University and Dr. Klar A at University of Cambridge. The authors would like to express the great appreciation to editors and reviewers for comments on this paper. References Addenbrooke, T. I., Potts, D. M., 2001. Twin tunnel interaction: surface and subsurface effects. International Journal of Geomechanics, ASCE 1 (2), 249-271.

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